EECS 2101 N, Winter 2024 – Assignment 4
Remember to write your full name and student number prominently on your submission. To avoid suspicions of plagiarism:
at the beginning of your submission, clearly state any resources (people, print, electronic) outside of the course
materials, and the course staff, that you consulted. You must submit a PDF file and Java files (if any) with the
required filename. The PDF file could be generated using L
A
T
E
X(recommended), Microsoft Word (saved as PDF, not the
.docx file), or other editor/tools. The submission must be typed. Handwritten submissions are not accepted.
Due April 15, 2024, 10:00 PM (on eClass); required file(s): a4sol.pdf
Answer each question completely, always justifying your claims and reasoning. Your solution will be graded not only on
correctness, but also on clarity. Answers that are technically correct that are hard to understand will not receive full marks.
Mark values for each question are contained in the [square brackets]. Your submitted file does not need to include/repeat
the questions — just write your solutions.
The assignment must be completed individually.
1. [10] We define the golden element of a collection of nelements to be the d0.618 ×ne-th smallest element of the collec-
tion (note: dxemeans the ceiling integer of x). For example, if the collection is [22, 72, 14, 14, 92, 80, 35, 22],
then n= 8 and d0.618 ×ne= 5, therefore the golden element is the 5th smallest element in the collection which is 35.
Note that duplicate elements are possible and each duplicate counts in the ranking — that’s why 35 is the 5th smallest
element rather than the 3rd.
Consider the following abstract data type that we will call a “GOLDEN-SET.”
Objects: A collection of elements that can be compared with each other. Duplicate elements are allowed (this is why
we use the term “collection” instead of “set”).
Operations:
•INSERT(S, x): Add element xto the collection S.
•FIND-GOLD(S): Return the golden element of collection Swithout removing it.
•DIG-GOLD(S): Remove and return the golden element of collection S.
Requirements: Let nbe the size of S,
•INSERT(S, x) must have worst-case runtime O(log n).
•FIND-GOLD(S) must have worst-case runtime O(1).
•DIG-GOLD(S) must have worst-case runtime O(log n).
In your PDF, describe your implementation of the GOLDEN-SET by completing the following parts.
(a) Describe what data structures you use to implement the GOLDEN-SET. You may use multiple data structures that
we learned in this course, and/or multiple instances of the same data structure. State clearly what are the struc-
tural properties/invariants that must be satisfied by these data structures (e.g., “the size of something must be
something”).
(b) For each of the above operations, provide the following information:
•A brief, high-level description of how the algorithm works for this operation.
•If the operation modifies the data (i.e., INSERT and DIG-GOLD), describe how the algorithm maintains the
structural property/invariant that you defined in Part (a).
•Analyze the worst-case runtime of the operation.
•Provide the pseudocode of the operation.
Note: Please do not repeat algorithms or runtime analyses from the lectures — just refer to known results as needed.
Hint: Consider using the heap data structure, and consider using more than one heaps.
